Optimal designing of structural topology is a problem to determine an optimal topology and shape size of structural members under a given condition. Hereinbelow, the structural member topology and shape size will be referred to as “design argument functions”, and the above decision problem will be referred to as a “design argument function optimization problem”. The term “argument function” is used since the topology and shape size are three-dimensional functions. In a design argument function optimization problem, an optimization problem of status argument function must be solved for values of each design argument function. From this meaning, the structural topology optimal designing can be regarded as a dual structure optimization problem having a status argument function optimization problem inside and a design argument function optimization problem outside.
In the inner status argument function optimization problem, the concept, based on accumulated technologies of dividing space into a finite number of elements is employed. Particularly in a problem with strain energy of structural member as an evaluation functional, a finite element method is generally applied as an analysis method. As a solution of finite element method, a direct method to linear equation is employed.
On the other hand, as to the design argument function optimization problem, briefly the following 3types of methods are provided (See non-patent document 1 or 2).    1. Evolutionary method (hereinbelow, “E method”)    2. Homogenization method (hereinbelow, “H method”)    3. Material distribution method (hereinbelow, “MD method) or Density method (hereinbelow, “D method”)
In the E method, each subspace obtained by space division is called a cell, and generation and deletion of cell is repeated in accordance with an appropriate rule. The structural members are given as a set of finally existing cells. As only two status, whether a cell exists or not, are permitted, clear structural members can be obtained. Further, as differential information of evaluation functional is not used, there is no trap in local optimal solution. Accordingly, the method is effective in a case where the evaluation functional is polymodal. In patent document 1, provided is a framed structural member optimization designing apparatus using a genetic search method which is a kind of the E method. The problem of conventional technical art, which has required trial computation based on accumulated know-how and which cannot be applied to an actual designing problem having a large number of design variables, is solved by the optimization designing apparatus, by the following arrangement. That is, an approximation optimization computer using an approximation equation for discrete design variable data such as frame member cross-sectional size, and a detailed optimization computer using the design variable data are provided, and these two computers are combined to a framed structural member optimization designing apparatus.
The H method enables use of sensitivity analysis by assuming a further fine structure as structural elements positioned in divided respective areas and introducing a design argument function taking a continuous value. The sensitivity analysis is an analysis method utilizing differential information of evaluation functional for a design argument function. If the sensitivity analysis is possible, an iterative solution such as a gradient method can be used. In comparison with a round-robin method such as the E method, at least computation time of search related to local optimal solution can be greatly reduced (See non-patent document 3).
The MD method or the D method represents changes in topology and shape size of structural members by allocating real numbers ranging from 0 to 1 indicating the rate of existence to respective structural members. These methods are similar to the H method in that the sensitivity analysis is enabled by replacing the discrete information as to whether or not a structural member exists with a continuous value of the rate of existence. However, as the number of parameters is smaller than that in the H method, the MD and the D method can be easily modeled and has a wide range of application.
The non-patent document 4 discloses a structure phase optimization method by the D method. The method has the following features. As a voxel finite element method (division of space at equal intervals) is employed, an element stiffness matrix is the same for every element. Accordingly, once the element stiffness matrix is computed, it can be used in subsequent computation. Further, as the elements are regularly arranged, it is not necessary to store nodal number information of the respective elements. As a conjugate gradient method with preconditioning and an element-by-element method are combined to solve a large scale simultaneous linear equations, a solution can be obtained without formation of global stiffness matrix. Thus the necessary memory capacity is small.
In the homogenization method, 6 design variables (in the case of three dimensional structure) are required for 1 element. Further, the element stiffness matrix must be re-calculated upon each change of design variable. On the other hand, if the density method to represent the rate of existence of structural number as a density ratio is employed, the number of design variables to 1 element is 1. In addition, the change of design variable does not influence the element stiffness matrix.
[Patent Document 1]
Japanese Patent Application Laid-open No. 11-314631
[Non-patent Document 1]
S. Bulman, J. Sienz and E. Hinton: “Comparisons between algorithms for structural topology optimization using a series of benchmark studies”, Computers and Structures, 79, pp. 1203-1218 (2001)
[Non-patent Document 2]
Y-L. Hsu, M-S. Hsu and C-T. Chen: “Interpreting results from topology optimization using density contours”, Computers and Structures, 79, pp. 1049-1058 (2001)
[Non-patent Document 3]
Hiroshi Yamakawa: “Optimization design”, Computational Mechanics and CAE series 9, Baifu-kan (1996)
[Non-patent Document 4]
Fujii, Suzuki and Ohtsubo: “Structure phase optimization using boxel finite element method”, Transactions of JSCES, Paper No. 20000010 (2000).
However, the above conventional techniques have the following drawbacks.
Generally, a structure optimization problem is formulated as a dual optimization problem including a status variable vector optimization problem at each repetitive step of design variable vector optimization problem. Assuming that the design variable vector optimization problem is referred to as an external optimization and the status variable vector optimization problem is referred to as an internal optimization, the internal optimization is a problem to obtain a status variable vector with a design variable vector as a parameter, i.e., the design variable vector being fixed. This problem is generally called structural analysis, which can be solved by using solution of linear equation with a finite element method.
However, if the structure has changed and a structural member in an area does not exist, e.g., a hole is formed in a member, the design variable corresponding to the element becomes 0. As a result, the Young's modulus of the element becomes 0. Then the coefficient matrix of the linear equation is not full-ranked, and the problem cannot be solved by a direct method since an inverse matrix cannot be calculated.
In the conventional art, when a design variable vector becomes 0, the following countermeasures are taken.
The equations are re-formed such that the coefficient matrix becomes full-ranked, i.e., the number of ranks and that of equations are equal.
Otherwise, the value of the design variable vector is replaced with a small value approximate to 0.
However, in the former method, as the equations may be updated upon each update of the design variable vector, it takes much computation time.
Further, in the latter method, setting the element design variable to a small value means that a physically thin film or a weak member exists. That is, in the conventional art, as a portion where material does not exist cannot be accurately represented, there is some doubt in the accuracy of obtained computation result.